Prolegomena to Any Future ‘Patapimpics

A naïve approach to ontology has led us to the apparently insoluble. Can a container be self-contained? In other words, is there a set of all sets? The assumption is that a set is any definable collection. This means sets can be defined in a loose manner.[18] This is seen in Cantor’s: P(x) “x is a cardinal number”.[19] This is the problem of unlimited comprehension that haunted Frege’s work.[20] You see, once parsed, the formal structure can be rendered: {x| x is a set}. This already anticipates the problem by being the problem. Indeed, Russell states the paradoxical nature of this in the following.[21]Let R = {x | x∉x} then R∈R⇔R ∉R.[22]This is the problem of intentionality and properties, although Russell was far from clear of this in his own formulations. In a similar way, Cantor slipped into the same error with the idea of an actual infinity, infinity as an object.[23] Assumptions must be made explicit and scrutinized if ontology is to be put on firm ground.

We’ll adopt Zermelo–Fraenkel axioms in order to satisfy the criteria of hereditary,[24] well-founded sets to account for the entities in our universe of discourse. This means we do away with all urelements because The COCK, like mathematical objects, is abstract,[25] and develop a formal apparatus.[26] To do this we must assume the empty set. Ø = { }. Nadda ex nihilo is Ø.[27] This is actually the beginning of cardinality. So, let’s start looking at οἱ λόγοι σπερματικοὶ.[28]

Ø = 0. If a ho has four teeth in her mouth {I I I I}[29] and I take a chain and lash her across the face, she now has zero teeth { }.[30] You see? 0 = Ø. Hit/miss.[31] Present/absent.[32]Fort/da. But not 0/1.[33] Not quite anyway, but, in binary terms, a qualified yes.[34] To get cardinals we need ordinals.[35] First we need the principle of extension. ∀A ∀B [∀x (x ∈ A ⇔x∈ B) ⇒ A = B].[36] If we are to have an iterative universe, this must be true. We could have ‘demonstrated’ this with any ‘set’, but we will look at it with the empty set.

In our iterative universe V we have elements to be considered, but most importantly we have Ø. If we make a set containing the empty set { Ø } we have a singleton. This is not the same thing. Ø ≠ { Ø }, but Ø ∈ { Ø }.[37] Excluding other elements, what we have actually done is to create the first two ranks of our universe V. At V0 we have Ø. At V2we have { Ø } We can make a further set from this a rank higher: { Ø, { Ø } } and then { Ø, { Ø, { Ø } } . This is ordinality. Now we can order ranks infinitely through nomination, but to look at how each set is comprised, or rather decomposed, in terms of its subsets, we need to consider the power set: ℘.[38]

The power set accounts for subsets. For any set A with n elements, it has 2n subsets. So, if V3 contains { Ø { Ø } } as an element then the power set contains { Ø, { Ø }, { { Ø } },{ Ø, { Ø } } as subsets.[39] From this we can see the iteration of sets. V0 = A, V1= A ∪ ℘(V0),[40]V2 = V1 ∪ ℘(V1), etc. Vω = Vω-1 ∪ ℘(Vω-1), etc. Vω + 1 = Vω ∪ ℘(Vω), etc. So do we approach infinity, or can we collect a set of all sets despite the paradox? If not a set, is there something that can group all sets? Container/contained? The von Neumann-Bernays alternative proposes the use of classes.[41] What this would mean is that class A would be at the level of Va which, to make it apparent in our hierarchical order, would make it a member of Va+1. However, again we should wield the Pimp Razor. As we have already proposed, the way to proceed is through Zermelo-Fraenkel axiomatics.[42]

Our earlier problem was that of Unlimited Comprehension. We want higher order sets that do not have the property that we are concerned with and that leads us into contradiction. What we want is exhaustibility, so we require the Axiom Schema of Separation,[43] traditionally rendered: ∀w1,…,wn∀A∃B∀x (x∈B⇔ [x∈A ∧ ϕ(x, w1,…, wn, A)]). This allowed a defined subset of a set to be a set in its own right and restricts set higher in the hierarchy.[44] We have already presupposed this above in our use of the formal use of the subset.[45] How about that? This is an aspect of pimpontology.

Now we can prove that there is not a set of all sets. Behold.

Suppose there is a set A of all sets. We will create a set not belonging to A.

B = {x ∈ A | x ∉ x}

Now, we assert B∉ A, and by the formation of B, so

B∈ B ⇔ (B ∈ A) ∧ (B∉ B).

If B ∈ A, then we have B ∈ B ⇔ B ∉ B

This is a contradiction, so B ∉ A

We now have a firm basis for our considerations, but we need to see how relations are supported. Ordered pairs <x,y> can be defined as follows {{x, {x, y}}. From this we have Cartesian coordinates, but this is actually a relation. A relation is not a graph or a substantive concept, it is a subset of a coordinate plane and, as such, it is the collection of alignments, thereby avoiding problematic ontological categories. This is an ontological reduction, and through set theoretic surrogates, we can operate with extension, and strictly extension, through n-ary relations. Now, considering this, we can see that any relation, any relation whatsoever, is a set of ordered pairs.[46] From here we can move to functions, identity relations and equivalence relations. So, now we have controlled inputs and outputs for one-to-one relations and onto relations with varying roots, so let us consider our Pimp universe and see how the set of sets is both actual, possible, and impossible.

We will begin with a basic relation.

Let set H be hoes and set B is bitches, if H ≠ ∅ and f: H–>B (where this is 1-1), then g: B–>H (where this is onto). So, let B’ = ran f, then there is a function y’ from B’ onto H. Consider f-1 = {<b,c>|<a,b> ∈f}

Since f-1 is 1-1, f-1 is a function

Let g’ = f-1, then g’ : B’–>H (where B’–>H is onto)

ran g’ = ran f-1 = dom f = H

Since H is non-empty, there is at least one a ∈ A

Dom u = B

i.e. g: B–> H, rany = H,

then there is a g:B–>H (where B–>H is onto)

In other words, all hoes are bitches. They participate in this space. However, that is not the end of the story. This is only an onto relation as hoes are always amongst other bitches.[47] However, in this pile of bitches are pimps as well.[48] We need to be able separate these pimps.[49] In order to do this, we still need a choice function for any non-empty set.[50]∀x [Ø∉ x⇒ ∃f: xà⋃x ∀A∈x (f(A) ∈ A)]. And here we go into ZFC, but not quite. It is ZFP because it is the pimp function that we are concerned with.[51] It is the pimp that makes selection and separation, but how? Remember, we have banished urelements, so these pimps, hoes, and bitches are formal. Just like numbers, we can identify them through another reduction of elemental relations.[52] So, if you’ve kept your eye on the ball, we are really talking about Pimp, Ho, and Bitch.[53]

The Pimp operates at the point of disjoint.[54] We have seen this. He is the maximal element of a set, and he is above the set,[55] but there is more. This is Tricky as it would seem we have fallen back into the fallacious reasoning that we have tried to avoid. We haven’t. This is the truth of Gz up Hoes down. You see, the belief in equinumerosity is where the flaw resides.[56] Von Neuman cardinality provides a simple alternative to Frege which allows numbers to be defined in terms of what preceded.[57] Likewise we can construct natural numbers,[58] and so forth. The problem comes when we consider infinity, the potential or the actual. Cantor had difficulty with actual infinity, but we need to assume it for our calculus. You see, there isn’t an axiomatic basis for actual infinity in our system, but we need to assert it inductively, otherwise our ability to compute will be severely hindered.[59] Assuming the provability of other functions,[60] let’s turn our full attention to the problem of infinity.

If we think of actual infinities, Cantor shows they are not all equal. The power set of any set is never equal to the set. This gives us an expanding hierarchy of cardinals, transfinite numbers (ℵ0, ℵ1, ℵ2, etc.). This takes us to the continuum hypothesis that would have the cardinality of the infinite cardinal series as the smallest uncountable cardinal number.[61] With the Axiom of Choice, Gödel found this consistent,[62] but he neglected the Pimp’s Lemmatic Schema. This lemma says yes and no. Limit and no limit. Why?

The Pimp’s Lemmatic Schema is the axiom that asserts the failure of axiomatics to totalize.[63] It is an answer to Hilbert’s second problem.[64] Primitive notions, axioms, and processes of iteration are for bitches. This is false foundationalism. The Pimp’s Lemmatic Schema reintroduces intensionality, disrupting theoretic surrogates, as the dialectic between applied and pure mathematics becomes suspect. Lines in the real world may not necessarily be describable in real numbers.[65] Consider the Everett interpretation and the quantization of series, an axiomatics rejecting axiomatics, and an improbably probability that does nothing for the hypothetico-reductive model.[66] This is not yet to mention the challenges on Euclidean space and our first intuitions about COCK. Let’s reconsider our axiomatics.

The Pimp’s Lemmatic Schema is a dubious lemma that permits and thwarts. It is the “yes, no, yes, no” of a drunken ho,[67] and for this we need to keep our eyes open.[68] It is a basic dialectic of in and out.[69] In the game, out of the game. In the system, out of the system.[70] We are already well acquainted with this interior/exterior. It is a consequence of Gödel’s metamathematics that shows our axioms inconsistent,[71] and there is no schema more inconsistent than the Pimp’s Lemmatic Schema since it is a product neither of theorems nor of logic as such.[72] How and why?

Undecidability becomes constituent to axiomatic undertakings. Attempting a metalanguage will not help, as it becomes impossible. Assumptions made about ontology are irrecoverably entangled in our epistemology. Language needs to be reconsidered as it is, for all the apparently formal purifying of ontology onto itself,[73] its exteriority is its interiority.[74] This is the Pimp’s Lemmatic Schema again, the pimp disjunct that names a ho by effacing her. The result of it is the Ho Function .

The Ho Function demonstrates that hoes are imaginary bitches.[75] You see, all hoes have an index of 2. This is because they are bitches compounded into an abstract state.[76] Some are more so than others, but the property is the same. Consider a ho2, a ho3, a ho4. They all equal the same thing. -1.[77] This is why our basic formulation is ἱ2 = -1. Mark the substitution.[78] Now, = ἱ and – ἱ. This is because of our directional rotation through the imaginary/ho dimension to spit her out as a bona fide ho. It is important to unpack this to see that (ἱ2 = -1) = (1 * ἱ * ἱ = -1). What is clear here is the role of polarized ones. 1 & -1. This involves a two-dimensional plane, or only apparently so. This is the Pimp’s Lemmatic Schema attained through the COCK Theorem. This would actually be in four dimensional space.

The COCK theorem asserts that the Pimp’s Lemmatic Schema operates in the three dimensional space of quaternions.[79] Here complex numbers are given planary representation over the field of real numbers. The pimp function would seem to operate on the 1/-1 of itself to ἱ2by way of the unit sphere R3. This hypercomplex number both removes and amplifies the valance depending on the modulation. Pimp’s Lemmatic Schema now applies itself to hoes from new angles. But there is a problem.

The use of quaternions would seem to be only the shadow of the shadow of COCK. Closer to its truth is use of tessarines.[80] Not readily susceptible to division, tessarines add another axis and give us another algebraic dimension. This would insert COCK, or rather something proximal to COCK, into our field. This further complicates our relations of pimp and ho. Let us consider what the implications are at the level of bitches.

The COCK seems to be felt through the Pimp’s Lemmatic Schema in the Ho Function . This theorem has the following transmogrifying structural effects. You see the HOLE is identified by COCK. This achieves the 1/-1 of the pimp matrix for the production of hoes. Remember, the world of pure bitches is a meaningless place. So, what we have is ( $ )–> $’ = -1, which really means $/$’ = -1. So, $’ = -1. This is the COCK theorem of COCK/Cock/cock or PIMP/Pimp/pimp, but their relation is maintained in a complex and erratic latticing.[81] What is clear, however, are the vectors. Like an armature, it serves as the basis for structuration.

This is actually tesseral as hoes are pieced together along with a world of meaning for bitches. This is how the Ho Function is triggered. This is the act of nomination,[82] but here, in our formulation is equi-vocaton as we have seen from the place of Bitch. This is our pimpontological algorithm in two different permutations. Each action challenges any possibility of a set of all sets by making a ho. This ho is hole in the symbolic network. Each ho is a hole as each pimp is a whole. This is the W/HOLE problem with COCK, inside and outside of meaning in the supra/infra bitch worlds.

The problem with our systematizing is that it is consistently inconsistent. This is a great strength, however we should realize that this because pimpontology is the handmaid of pimpology. Pimpontology should secure pimpology, and it does, but it does so both reflexively and impossibly.[83] Again, this is not a weakness, but the question is how do they relate? Pimpology is typically descriptive as it expands willy-nilly, but how does this affect its pimpontological basis? This would seem incorrect. It isn’t. This is the second prong of COCK’s riddle and mystery. What has happened is that language has been dragged kicking and screaming out of the front door by bitches and let in the back door by liveried pimps. This is why.

Pimpontology is, by definition, formal. It abstracts while trying to deal with abstractions for those abstractions to make sense in our world of application. A simplification, yes, but it finds itself in a type of hermeneutic circle.[84] Pretending language and pimpology is extraneous is a stance for bitches. Russell of the flawed and abandoned Theory of Knowledge tried to account for the logical and perceptible, relations and sense data.[85] Logic and epistemology still cannot be aligned as seen in problems with judgment of the levels of propositional relations.[86] Improved upon was a more developed logical atomism,[87] but universals and the Platonic legacy persisted. Wittgenstein of the Tractatus-Logico Philosophicus moved to further logical deduction with picture theory.[88] However, pimpology and paralanguage deals more with the Wittgenstein of Philosophical Investigations.[89] Let us continue and reconsider our position.

[1] This section was completed with the help of Tommy the Motherfucking Autist. Using Dazzle Razzle’s notes, inputs were fed into Tommy’s head and much of the resultant data was able to be synthesized in a way consonant with theorizations and finding in Dazzle Razzle’s documents. However, Tommy is handicapped and, like Donna Inez, “[his] thoughts were theorems, [his] words a problem.”

[2] This footnote is a non sequitur, but this chapter should have been called Non Sequuntur, which, upon closer examination, you will find somewhat circular.

[3] Not an unproblematic statement. It is plagued with issues of identity and integrity, and regression for that matter.

[4] Would Abelard be considered a Nominalist if universals were able to receive predication? Good question. Perhaps we are anticipating the challenges of sets and classes.

[5] There are variant forms of everything listed above, but we’ll settle for a simplification.

[6] As you are likely able to infer, Dazzleans actually favor Iamblichus as an antecedent.

[15]I.e., numbers and functions. You know, what allowed Wittgenstein to jerk off in the trenches.

[16] Indeed, ordine geometrico demonstrate. However, along with Frege, our approach will be fully logical. The actual ‘spatial’ assumptions of geometry do not apply. Fuck what ya heard. It is the methodology that is of interest.

[17] We will omit many axioms due to tedium. You will need to fill in more than a few blanks.

[20] Frege’s formalism could express “Every boy loves some girl who loves some boy who loves every boy.” But he couldn’t achieve the nuances of Damon Albarn who was able to express, “Girls who are boys who like boys to be girls who do boys like they’re girls who do girls like they’re boys.” It’s all about expression, and Frege spoke German. Du bist sehr schon is very different than Du bist sehr schön. It is true, but Frege also liked formal languages and this pimpnote should really be in regard to Basic Law V, ε’ƒ(ε) = α’g(α) ≡ ∀x[ƒ(x) = g(x)], but oh well. Also, assume that the apostrophe (or smooth breathing aspiration mark, if you prefer) is over the epsilon and alpha, okay? It might be better to just switch it with Hume’s principle anyway.

[21] We assume all basic definitions and logical operators as established.

[22] Similarly, Russell saw the problem of sets that contain themselves as elements: ϕ(x) to be ¬(x ∊ x). To circumvent this, Russell introduced the not unproblematic theory of types. Well, Frege, an inveterate Platonist, will always have the third realm.

[23] Cantor conflated finite sets as objects with infinite sets. This led to transfinite numbers beginning with א0when the infinite set of natural numbers is considered. This then is not ‘finite’ as it considers totalities, it appreciate their inconsistence, their failure to totalize.

[25]Or rather, is not ab-stracted. Admittedly, the above use is shamefully solecistical.

[26]I.e., we will not assume the ontology of things, people, etc. Nor will we consider classes, while direct sets can only receive indirect treatment, for that matter. Within these axiomatics we are strictly concerned with sets. This is not a limitation as you will see.

[32] This is a truth of Chain Fight Wisdom. Once I show you the chain, it means you have no teeth. If I don’t show you the chain, it means you have teeth. It turns the present/absent dialectic on its head, so to speak. See Intermezzo: How to Be a Motherfucker for more on Chain Fight Wisdom.

[33] Ironically there is only one Ø, and this is a consequence of extensionality. Yes and no, it is always there, present in every set, and it isn’t.

[34] We will see the iterability of the binary system later. It depends on the same empty set dialectic as what we are pursuing here.

[35] Not exactly. By von Neumann cardinal assignment, the empty set supplies the cardinal 0. A lot hinges on this, and etymology supports it.

[36] This allows us to say that if A and B are sets, the for every object x, x∈ A iff x∈ B. then we can affirm that A = B

[44] Austere English help facilitate this quality of defining and disambiguating that status of would-be properties through precision and exactitude in a quasi-algebra. Effectively it is first-order logic with identity and membership relations.

[45] Heads up, we already assumed the axiom of intersection, union, difference…ZF has ten axioms. We will encounter four more.

[47] Tricks, gangstas, transvestites, etc. Also you will have noted that H ≠ ∅. This is true, but it is because it is not an equivalence relation. Hoes are ∅ via the vagina. That cannot be expressed in this notation. One requires the Pimp’s Lemmatic Schema.

[48] A needle in a haystack is an adequate analogy, but with a twist. A pimp is like a needle that makes itself a needle, makes you roll in the hay, and charges you for it.

[49] Because pimp, represented as $’, can be expressed in the following relation. $’≾ B, as ran f, but ran f ⊀ $’. This is because it happens at a different space.

[50] This is not a first-order concern. This is what allows sets of functions to be possible, to be grouped as a single choice, etc. Choice as such needs no axiom.

[51] The Pimp’s Lemmatic Schema is an interesting one as it both paradoxically supplants and supplements the Axiom of Choice. You will see the connection if you consider Hilbert, Bourbaki, and Sokal notation.

[52]This has already been done through theoretic surrogates. All abstract objects can be reduced to sets.

[53] No need to demure. We know that PIMP, HO, and BITCH (viz. PIMP) are pulling the strings in the complicated way that we have already seen. However, we need to know what role COCK plays.

[55] The former is a quality of (Max “this is similar to the Axiom of Choice” Zorn) Zorn’s Lemma, the later of (Frampton’s) Zorn’s Lemma as sets can only be completed in by decompletion.

[56] Not just the criterion of cardinal number of a set being infinite iff it is equinumerous with at least one of its own proper subsets, but the failure of gauging infinites. For instance, |ℝ| > |ℕ| because there is no one-to-one, onto function, so |ℝ| ≠ |ℕ|. The proof is in Cantor’s diagonalization.

[61] Really what we are saying is |ℝ| = 2ℵ0, and, if so, how many infinite cardinals are there between ℵ0 and 2ℵ0 ? You’ll see that the aspects of COCK are analogous to inaccessible cardinals. *Editorial note* The ‘0’ should be a subscripted superscript. There have been ongoing problems with notation here. Tommy the Motherfucking Autist has be chastised. If he does not learn, he will be put back in his cage.

[63] In this way it is similar to Gödel’s second incompleteness theorem. For Robinson arithmetic to be recursively axiomatized, the axiomatic system used needs to be inconsistent to show its consistency.

[64] Not that Gödel’s second incompleteness theorem didn’t, just that Pimp’s Lemmatic Schema is better.

[70] All of these are ontologically problematic, but are (in)consistent through pimpontology.

[71] Actually, the second incompleteness theorem states that if our axioms are consistent, then we cannot use them to support our model or our axioms. So the model is possible, but not provable. Thank goodness for the Pimp’s Lemmatic Schema!

[72] Because it is pimpontological. For instance, the Pimp’s Lemmatic Schema gets angry when surjective efforts are made through the axiom schema of replacement. This smacks too much of classes and fails by expressing its own consistency. Forget Quine’s universal set as well.

[78] Instead of ho2 we have ἱ2. Instead of an italicized ‘h’ or any other roman letter, we have used the rough aspirated iota. The ‘Hhhh’ is for ‘ho’. The iota is of a googological function that terminates in 0. Hence Hhh0. Cash is a sequence, but it ends in 0 when her term of service is complete.

[88] At this point his convictions were that the problems of traditional philosophy are to be found in an ignorance of symbolism and borne out in the misuse of language. For this perhaps we should turn to the triadic nature of Piercean semiotics to align pimpology and pimpontology.

[89] As Socrates of the Phaedo says, “I decided to take refuge in language, and study the truth of things by means of it.”